1,370 research outputs found
Computing Dynamic Output Feedback Laws
The pole placement problem asks to find laws to feed the output of a plant
governed by a linear system of differential equations back to the input of the
plant so that the resulting closed-loop system has a desired set of
eigenvalues. Converting this problem into a question of enumerative geometry,
efficient numerical homotopy algorithms to solve this problem for general
Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While
dynamic feedback laws offer a wider range of use, the realization of the output
of the numerical homotopies as a machine to control the plant in the time
domain has not been addressed before. In this paper we present symbolic-numeric
algorithms to turn the solution to the question of enumerative geometry into a
useful control feedback machine. We report on numerical experiments with our
publicly available software and illustrate its application on various control
problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly
available via http://www.math.uic.edu/~jan/download.htm
A non-linear structure preserving matrix method for the low rank approximation of the Sylvester resultant matrix
A non-linear structure preserving matrix method for the computation of a structured low rank approximation S((f) over bar , (g) over bar) of the Sylvester resultant matrix S(f , g) of two inexact polynomials f = f(y) and g = g(y) is considered in this paper. It is shown that considerably improved results are obtained when f (y) and g(y) are processed prior to the computation of S((f) over bar , (g) over bar), and that these preprocessing operations introduce two parameters. These parameters can either be held constant during the computation of S((f) over bar , (g) over bar), which leads to a linear structure preserving matrix method, or they can be incremented during the computation of S((f) over bar, (g) over bar), which leads to a non-linear structure preserving matrix method. It is shown that the non-linear method yields a better structured low rank approximation of S((f) over bar , (g) over bar) and that the assignment of f (y) and g(y) is important because S((f) over bar , (g) over bar) may be a good structured low rank approximation of S(f, g), but S((f) over bar , (g) over bar) may be a poor structured low rank approximation of S (g f) because its numerical rank is not defined. Examples that illustrate the differences between the linear and non-linear structure preserving matrix methods, and the importance of the assignment off (y) and g(y), are shown. (C) 2010 Elsevier B.V. All rights reserved
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Nearest common root of a set of polynomials: A structured singular value approach
The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations
The computation of the degree of an approximate greatest common divisor of two Bernstein polynomials
This paper considers the computation of the degree t of an approximate greatest
common divisor d(y) of two Bernstein polynomials f(y) and g(y), which are
of degrees m and n respectively. The value of t is computed from the QR
decomposition of the Sylvester resultant matrix S(f, g) and its subresultant
matrices Sk(f, g), k = 2, . . . , min(m, n), where S1(f, g) = S(f, g). It is shown
that the computation of t is significantly more complicated than its equivalent
for two power basis polynomials because (a) Sk(f, g) can be written in several
forms that differ in the complexity of the computation of their entries, (b)
different forms of Sk(f, g) may yield different values of t, and (c) the binomial
terms in the entries of Sk(f, g) may cause the ratio of its entry of maximum
magnitude to its entry of minimum magnitude to be large, which may lead to
numerical problems. It is shown that the QR decomposition and singular value
decomposition (SVD) of the Sylvester matrix and its subresultant matrices yield
better results than the SVD of the B´ezout matrix, and that f(y) and g(y)
must be processed before computations are performed on these resultant and
subresultant matrices in order to obtain good results
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
Matrix representation of the shifting operation and numerical properties of the ERES method for computing the greatest common divisor of sets of many polynomials
The Extended-Row-Equivalence and Shifting (ERES) method is a matrix-based method developed for the computation of the greatest common divisor (GCD) of sets of many polynomials. In this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the ERES method by introducing its complete algebraic representation. Then, we analyse in depth its overall numerical stability in finite precision arithmetic. Numerical examples and comparison with other methods are also presented
Comparison of algorithms for calculation of the greatest common divisor of several polynomials
summary:The computation of the greatest common divisor (GCD) has many applications in several disciplines including computer graphics, image deblurring problem or computing multiple roots of inexact polynomials. In this paper, Sylvester and BĂ©zout matrices are considered for this purpose. The computation is divided into three stages. A rank revealing method is shortly mentioned in the first one and then the algorithms for calculation of an approximation of GCD are formulated. In the final stage the coefficients are improved using Gauss-Newton method. Numerical results show the efficiency of proposed last two stages
The Orevkov invariant of an affine plane curve
We show that although the fundamental group of the complement of an algebraic
affine plane curve is not easy to compute, it possesses a more accessible
quotient, which we call the Orevkov invariant.Comment: 20 page
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